Rethinking Set Theory
نویسنده
چکیده
Mathematicians manipulate sets with confidence almost every day of their working lives. We do so whenever we work with sets of real or complex numbers, or with vector spaces, topological spaces, groups, or any of the many other set-based structures. These underlying set-theoretic manipulations are so automatic that we seldom give them a thought, and it is rare that we make mistakes in what we do with sets. However, very few mathematicians could accurately quote what are often referred to as ‘the’ axioms of set theory. We would not dream of working with, say, Lie algebras without first learning the axioms. Yet many of us will go our whole lives without learning ‘the’ axioms for sets, with no harm to the accuracy of our work. This suggests that we all carry around with us, more or less subconsciously, a reliable body of operating principles that we use when manipulating sets. What if we were to write down some of these principles and adopt them as our axioms for sets? The message of this article is that this can be done, in a simple, practical way. We describe an axiomatization due to F. William Lawvere [3, 4], informally summarized in Fig. 1. The axioms suffice for very nearly everything mathematicians ever do with sets. So we can, if we want, abandon the classical axioms entirely and use these instead.
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عنوان ژورنال:
- The American Mathematical Monthly
دوره 121 شماره
صفحات -
تاریخ انتشار 2014